In this paper, we are proposing certain modifications of Szász-Mirakian type operators and study their approximation properties. We also give a Voronovskaya type theorem for these operators.
Mathematics Subject Classification (2010): 41A25, 41A36.
Received 20 April 2022; Accepted 14 July 202

AGRAWAL, Raksha Rani

GUPTA, Nandita

English

Studia Universitatis Babeş-Bolyai

Stud. Univ. Babeş-Bolyai Math. 69(2024), No. 2, 415–424DOI: 10.24193/subbmath.2024.2.11Generalized Szász-Mirakian type operatorsRaksha Rani Agrawal and Nandita GuptaAbstract. In this paper we propose certain modifications of Szász-Mirakian typeoperators and study their approximation properties. We also give a Voronovskayatype theorem for these operators.Mathematics Subject Classification (2010): 41A25, 41A36.Keywords: Linear positive operators, Schurer type operator, Szász-Mirakian op-erators, Voronovskaya type theorem.1. IntroductionClassical Szász-Mirakian operator is defined asSn(f ;x) = e−nx∞∑k=0(nx)k(k)!f(kn)(1.1)with x ∈ R0 = [0,∞), n ∈ N = {1, 2, 3, . . . }, k ∈ N0 = N ∪ 0 and f ∈ C(R0). Theapproximation behaviour of this operator for bounded functions is well known (see,e.g. [2, 8]). Hermann considered this operator on a much wider class, growing fasterthan exponentially. In 2005, Schurer[6, 7] type generalization was given by Moreno[4] for this operator 1.1.Sn,p(f ;x) = e−(n+p)x∞∑k=0((n+ p)x)k(k)!f(kn), x ∈ R0, n ∈ N, p ∈ N0.Later Firlej and Rempulska [3] introduced a modified Szász-Mirakian operator:S̄n(f ;x) =1cosh(nx)∞∑k=0(nx)2k(2k)!f(2kn), x ∈ R0, n ∈ N.Received 20 April 2022; Accepted 14 July 2023.© Studia UBB MATHEMATICA. Published by Babeş-Bolyai UniversityThis work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives4.0 International License.416 Raksha Rani Agrawal and Nandita GuptaMotivated by the above two modifications now we consider Szász-Mirakian type opera-tors for f ∈ CBŜn,p(f ;x) =1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!f(2kn), x ∈ R0, n ∈ N, p ∈ N0. (1.2)and coshx, tanhx are elementary hyperbolic functions. LetC2B = {f ∈ CB ∩ C2(R0) : f ′; f ′′ ∈ CB}be the space of real-valued functions uniformly continuous and bounded onR0 = [0;∞) and let the norm in CB be given by the formula‖f‖ = supx∈R0|f(x)|.In the year 2008, Deo et al. [1] studied Voronovskaya type results for modified Bern-stein operators. Now the purpose of this study is to give Voronovskaya type theo-rems for Schurer ([6, 7]) as well Firlej and Rempulska [3] type modification of Szász-Mirakian operators.2. Auxiliary resultsIn this section we prove some results on Ŝn,p that will help in establishing themain result.Lemma 2.1. For each n ∈ N and x ∈ R0 we haveŜn,p(1;x) = 1, (2.1)Ŝn,p(t;x) =(n+ p)xntanh((n+ p)x), (2.2)Ŝn,p(t2;x) =((n+ p)x)2n2+(n+ p)xn2tanh((n+ p)x), (2.3)Ŝn,p(t3;x) =1n3[{((n+ p)x)3 + (n+ p)x} tanh((n+ p)x) + 3((n+ p)x)2], (2.4)andŜn,p(t4;x) =1n4[((n+p)x)4 +{6((n+p)x)3 +(n+p)x} tanh((n+p)x)+7((n+p)x)2].(2.5)Proof. From (1.2) we can easily obtain (2.1) andŜn,p(t;x) =1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!(2kn)=(n+ p)xntanh((n+ p)x).Generalized Szász-Mirakian type operators 417Ŝn,p(t2;x) =1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!(2kn)2=1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!{(2k)(2k − 1) + 2k}n2=((n+ p)x)2n2+(n+ p)xn2tanh((n+ p)x)Ŝn,p(t3;x) =1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!(2kn)3=1cosh((n+ p)x)∞∑k=0((n+p)x)2k(2k)!{(2k)(2k−1)(2k−2) + 6k(2k−1) + 2k}n3=1cosh((n+ p)x)[ ((n+ p)x)3 + (n+ p)xn3sinh((n+ p)x)+3((n+ p)x)2n3cosh((n+ p)x)]=1n3[((n+p)x)3 tanh((n+p)x) + 3((n+p)x)2 + (n+p)x tanh((n+p)x)].Ŝn,p(t4;x) =1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!(2kn)4=1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!n4[(2k)(2k − 1)(2k − 2)(2k − 3)+ 6(2k)(2k − 1)(2k − 2) + 7(2k)(2k − 1) + 2k]=1n4[((n+p)x)4 + (6((n+p)x)3 + (n+p)x) tanh((n+p)x) + 7((n+p)x)2].Using above Lemma 2.1, we shall prove the following Lemma.Lemma 2.2. The following equalities hold for all x ∈ R0 and n ∈ N :Ŝn,p(t− x;x) =(n+ p)xn[tanh((n+ p)x)− 1] + pxnŜn,p((t− x)2;x) =(n+ p)n(2x2 − xn)(1− tanh(n+ p)x) + (n+ p)x+ p2x2n2Ŝn,p((t− x)3;x) =(n+ p)ntanh((n+ p)x)[(n+ p)2n2x3 +xn2− 3x2n+ 3x3]+3(n+ p)2x2n3− 3x(n+ p)2x2n2− x3418 Raksha Rani Agrawal and Nandita GuptaŜn,p((t− x)4;x)=(n+ p)ntanh((n+ p)x)[6(n+ p)2x3n3− 4(n+ p)2x3n2+xn3− 4x2n2+6x3n− 4x4]+((n+ p)x)4 + 7((n+ p)x)2 − 12xn((n+ p)x)2 + 6x2n2((n+ p)x)2 + x4n4n4Proof. Using Lemma 2.1 we have,Ŝn,p(t− x;x) =(n+ p)xntanh((n+ p)x)− x=(n+ p)xn[tanh((n+ p)x)− 1] + pxn(2.6)Ŝn,p((t− x)2;x) =((n+ p)x)2n2+(n+ p)xn2tanh((n+ p)x)− 2x (n+ p)xntanh((n+ p)x) + x2= tanh((n+ p)x)[(n+ p)xn2− 2x (n+ p)xn]+ x2 +((n+ p)x)2n2(2.7)=(n+ p)n(2x2 − xn)(1− tanh(n+ p)x) + (n+ p)x+ p2x2n2Ŝn,p((t− x)3;x)=1n3[((n+ p)x)3 tanh((n+ p)x) + 3((n+ p)x)2 + (n+ p)x tanh((n+ p)x)]− 3x[((n+ p)x)2n2+(n+ p)xn2tanh((n+ p)x)]+ 3(n+ p)x3ntanh((n+ p)x)− x3 (2.8)=(n+ p)ntanh((n+ p)x)[(n+ p)2n2x3 +xn2− 3x2n+ 3x3]+3(n+ p)2x2n3− 3x(n+ p)2x2n2− x3Ŝn,p((t−x)4;x)= Ŝn,p(t4;x)−4xŜn,p(t3;x)+6x2Ŝn,p(t2;x)−4x3Ŝn,p(t;x)+x4Ŝn,p(1;x)= tanh((n+ p)x)[ (n+ p)x+ 6((n+ p)x)3n4− 4x{((n+ p)x)3 + (n+ p)x}n3+6x3(n+ p)− 4x4n(n+ p)n2]+((n+ p)x)4 + 7((n+ p)x)2 − 12xn((n+ p)x)2 + 6x2n2((n+ p)x)2 + x4n4n4=(n+ p)ntanh((n+ p)x)[6(n+ p)2x3n3− 4(n+ p)2x3n2+xn3− 4x2n2+6x3n− 4x4]Generalized Szász-Mirakian type operators 419+((n+ p)x)4 + 7((n+ p)x)2 − 12xn((n+ p)x)2 + 6x2n2((n+ p)x)2 + x4n4n4(2.9)In order to prove a Voronovskaya type theorem we need the following lemmas.Lemma 2.3. For n, r ∈ N, p ∈ N0 and x ≥ 0, the following results hold(a) 0 ≤ xr(1− tanh(n+ p)x) ≤ 21−rr!(n+ p)−r(b) limn→∞[n {tanh((n+ p)x)− 1}] = 0(c) limn→∞{tanh((n+ p)x)} = 1,and(d) limn→∞{tanh((n+ p)x)− 1}n= 0Proof. We shall use an inequality considered in (eq.(22), [5]) which says that, form, r ∈ N and x ≥ 0,0 ≤ xr(1− tanhmx) ≤ 21−rr!m−rNow on replacing m by n+ p, n ∈ N, p ∈ N0 we get the desired result.Following the technique used in Lemma 2 of [5],we easily obtain (b),(c) and (d). Lemma 2.4. For every fixed x ∈ R0, we have(i) limn→∞nŜn,p(t− x;x) = px,(ii) limn→∞nŜn,p((t− x)2;x) = x,Proof. Using Lemma 2.2 we obtain,(i)Ŝn,p(t− x;x) =(n+ p)xntanh((n+ p)x)− x=(n+ p)xn[tanh((n+ p)x)− 1] + pxn= x [tanh((n+ p)x)− 1] + pxntanh((n+ p)x).thereforenŜn,p(t− x;x) = xn [tanh((n+ p)x)− 1] + px tanh((n+ p)x).Using Lemma 2.3 we getlimn→∞nŜn,p(t− x;x) = limn→∞[xn {tanh((n+ p)x)− 1}+ px tanh((n+ p)x)]= px.(ii)Ŝn,p((t− x)2;x) =(n+ p)n(2x2 − xn)(1− tanh(n+ p)x) + (n+ p)x+ p2x2n2420 Raksha Rani Agrawal and Nandita GuptanŜn,p((t− x)2;x) = 2x2[n {1− tanh((n+ p)x)}]+ 2px2 [1− {tanh((n+ p)x)}]− x [{1− tanh((n+ p)x)}]− pxn[1− {tanh((n+ p)x)}] + x+ pxn+p2x2nAgain using Lemma 2.3 we obtainlimn→∞nŜn,p((t− x)2;x) = x Lemma 2.5. The following inequalities are satisfied for all n ∈ N, p ∈ N0 and x ∈ R0 :|Ŝn,p(t− x;x)| ≤px+ 1n|Ŝn,p((t− x)2;x)| ≤3 + (p+ 1)x+ p2x2n|Ŝn,p((t− x)4;x)| ≤47 + (1 + p)x+ (3 + 10p+ 7p2)x2 + 6p2x3 + p4x4n2(2.10)Proof. For n, r ∈ N, p ∈ N0 and x ≥ 0 we have0 ≤ xr(1− tanh(n+ p)x) ≤ 21−rr!(n+ p)−r.So from Lemma 2.2,|Ŝn,p(t− x;x)| =∣∣∣∣ (n+ p)xn [tanh((n+ p)x)− 1]∣∣∣∣+ pxn≤ px+ 1nŜn,p((t− x)2;x) =((n+ p)x)2n2+(n+ p)xn2tanh((n+ p)x)− 2x (n+ p)xntanh((n+ p)x) + x2= tanh((n+ p)x)[(n+ p)xn2− 2x (n+ p)xn]+ x2 +((n+ p)x)2n2= (tanh(n+ p)x− 1)[(n+ p)x)n2− 2x (n+ p)xn]+(n+ p)x+ p2x2n2|Ŝn,p((t− x)2;x)| =∣∣∣∣(tanh(n+p)x−1) [ (n+p)x)n2 − 2x (n+p)xn]+(n+ p)x+ p2x2n2∣∣∣∣≤ 1 + (n+ p)x+ p2x2n2+2n(n+ p)≤ 3 + px+ p2x2n2+xn≤ 3 + px+ x+ p2x2n≤ 3 + (p+ 1)x+ p2x2nGeneralized Szász-Mirakian type operators 421Ŝn,p((t− x)4;x)=(n+ p)ntanh((n+ p)x)[6(n+ p)2x3n3− 4(n+ p)2x3n2+xn3− 4x2n2+6x3n− 4x4]+((n+ p)x)4 + 7((n+ p)x)2 − 12xn((n+ p)x)2 + 6x2n2((n+ p)x)2 + x4n4n4= tanh((n+p)x)[6(n+p)3x3n4− 4(n+p)3x4n3+(n+p)xn4− 4(n+ p)x2n3+6(n+ p)x3n2− 4(n+ p)nx4]+(n+ p)4x4n4+7(n+ p)2x2n4− 12(n+ p)2x3n3+6(n+ p)2x4n2+ x4=(tanh((n+ p)x)− 1)[6(n+ p)3x3n4− 4(n+ p)3x4n3+(n+ p)xn4− 4(n+ p)x2n3+6(n+p)x3n2− 4(n+p)nx4]+((n+p)4n4− 4(n+p)3n3+6(n+ p)2n2− 4(n+ p)n+ 1)x4+(6(n+ p)3n4+6(n+ p)n2− 12(n+ p)2n3)x3 +(7(n+ p)2n4− 4(n+ p)n3)x2 +(n+ p)n4x|Ŝn,p((t− x)4;x)| ≤10n4+16n3(n+ p)+9n2(n+ p)2+12n(n+ p)3+p4x4n4+6p2x3n3+(3 + 10p+ 7p2)x2n2+(1 + p)xn2≤ 47n4+p4x4 + 6p2x3 + (3 + 10p+ 7p2)x2 + (1 + p)xn2≤ 47 + (1 + p)x+ (3 + 10p+ 7p2)x2 + 6p2x3 + p4x4n23. Voronovskaya type theoremsIn this section we give a Voronovskaya-type theorem for the operators Ŝn,p withthe help of properties of Ŝn,p, which are already mentioned in the above lemmas.Lemma 3.1. Suppose that x0 is a fixed point in R0 and ϕ(t;x0)is a given functionbelonging to CB and such thatlimt→x0ϕ(t;x0) = 0(limt→pϕ(t; 0) = 0)Then for a fixed p ∈ Nlimn→∞Ŝn,p(ϕ(t;x0);x0) = 0. (3.1)Proof. By (1.2) we have for n ∈ N and a fixed point x0 ≥ 0Ŝn,p(ϕ(t;x0);x0) =1cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!ϕ(2kn;x0)(3.2)422 Raksha Rani Agrawal and Nandita GuptaChoose ε > 0. Since ϕ(·;x0) ∈ CB , there exists a positive constant δ ≡ δ(ε) such that|ϕ(t;x0)| <ε2, if |t− x0| < δ, t ≥ 0Moreover there exists a positive constant M such that |ϕ(t;x0)| ≤ M for all t > 0.Hence, from (3.2) we get for every n ∈ N|Ŝn,p(ϕ(t;x0);x0)| ≤1cosh((n+ p)x)∑k∈Q1,n((n+ p)x)2k(2k)!∣∣∣∣ϕ(2kn ;x0)∣∣∣∣+1cosh((n+ p)x)∑k∈Q2,n((n+ p)x)2k(2k)!∣∣∣∣ϕ(2kn ;x0)∣∣∣∣= E1 + E2 (3.3)whereQ1,n = k ∈ N0 :∣∣∣∣2kn − x0∣∣∣∣ < δandQ2,n = k ∈ N0 :∣∣∣∣2kn − x0∣∣∣∣ ≥ δ.From (2.1) we get,E1 <ε21cosh((n+ p)x)∞∑k=0((n+ p)x)2k(2k)!=ε2(3.4)E2 ≤M1cosh((n+ p)x)∑k∈Q2,n((n+ p)x)2k(2k)!(3.5)Since | 2kn − x0| ≥ δ implies 1 ≤ δ−2 ( 2kn − x0)2, we can writeE2 ≤ Mδ−21cosh((n+ p)x))∑k∈Q2,n((n+ p)x)2k(2k)!(2kn− x0)2≤Mδ−2Ŝn,p((t− x0)2;x0)which by Lemma 2.5 gives,E2 ≤M(3 + px0 + x0 + p2x20)nδ2It is obvious that for given ε > 0, δ > 0,M > 0 and x0 ≥ 0 we can choose n0 ≡n0(ε; δ;M ;x0) ∈ N such that for all natural numbers n > n0 one getsM(3 + px0 + x0 + p2x20)nδ2<ε2Hence,E2 <ε2for n > n0. (3.6)Using equations (3.4) and(3.6) to (3.3) we getlimn→∞Ŝn,p(ϕ(t;x0);x0) = 0Generalized Szász-Mirakian type operators 423and this completes the proof. Theorem 3.2. If f ∈ C2B, then for every fixed x ∈ R0 one getslimn→∞n{Ŝn,p(f ;x)− f(x)} = pxf ′(x) +x2f ′′(x). (3.7)Proof. Let x0 ∈ R0 be a fixed point.Then by Taylor formula we can write for everyt ∈ R0,f(t) = f(x0) + f′(x0)(t− x0) +12f ′′(x0)(t− x0)2 + ψ(t;x0)(t− x0)2, (3.8)where ψ(t;x0) ∈ CB and limt→x0ψ(t;x0) =(limt→0+ψ(t; 0) = 0)On applying the operator Ŝn,p on both sides of (3.8), we have for every n ∈ N ,Ŝn,p(f ;x0)− f(x0) = f ′(x0)Ŝn,p(t− x0;x0) +12f ′′(x0)Ŝn,p((t− x0)2;x0)+ Ŝn,p(ψ(t;x0)(t− x0)2;x0) (3.9)In view of (3.9) and by Holder’s inequality we get for n ∈ N|Ŝn,p(ψ(t;x0)(t− x0)2;x0)| ≤{Ŝn,p(ψ2(t;x0);x0)}1/2 {Ŝn,p((t− x0)4;x0)}1/2(3.10)Since the function ϕ(t;x0) = ψ2(t;x0), t ≥ 0 satisfies the assumption of Lemma 3.1we have,limn→∞Ŝn,p(ψ2(t;x0);x0) = 0From (2.10) it follows that there exists a constant M1 = M1(p, x0) depending onp and x0 such thatn2Ŝn,p((t− x0)4;x0) ≤M1 for n ∈ NConsequently we obtainlimn→∞nŜn,p(ψ(t;x0)(t− x0)2;x0) = 0 (3.11)from (3.10). Using Lemma 2.4 and (3.11), we derive immediately (3.7) from (3.9).Thus the proof is complete. References[1] Deo, N., Noor, M.A., Siddiqui, M.A., On approximation by a class of new Bernsteintype operators, Appl. Math. Comput., 201(2008), 604-612.[2] Ditzian, Z., Totik, V., Moduli of Smoothness, Springer-Verlag, New York, 1987.[3] Firlej, B., Rempulska, L., Approximation function by some linear positive operators,Fasciculi Mathematici, 27(1997), 65-79.[4] Lopez-Moreno, A.J., Preservation of Lipschitz constants by Bernstein type operators,Acta Math. Hungar., 102(4)(2005), 343-354.[5] Rempulska, L., Skorupa, M., A Voronowskaja-type theorem for some linear positive op-erators, Indian Journal of Mathematics, 39(1997), no. 2, 127-137.424 Raksha Rani Agrawal and Nandita Gupta[6] Schurer, F., Linear positive operators in approximation theory, Math. Inst. Techn. Univ.Delft Report, (1962).[7] Schurer, F., On Linear Positive Operators in Approximation Theory, Dissertation, Uni-versity of Delft, 1965.[8] Totik, V., Approximation by Bernstein polynomials, Amer. J. Math., 116(1994), 995-1008.Raksha Rani AgrawalDepartment of Mathematics,S.S.T.C./S.S.G.I. College,Durg, Chhattisgarh, Indiae-mail: raksha mukesh@yahoo.comNandita GuptaGovernment Polytechnic,Mahasamund, Chhattisgarh, Indiae-mail: nandita dec@yahoo.com

Generalized Szász-Mirakian Type Operators

https://studia.reviste.ubbcluj.ro/index.php/subbmathematica/article/download/7303/7082

Generalized Szász-Mirakian Type Operators

Abstract

Similar works

Full text

Available Versions

Studia Universitatis Babeş-Bolyai